What a ratio says, how to keep it the same while the numbers grow, and the one division that turns any rate into a unit rate — with tables, tape diagrams and double number lines, the better buy, speed, percent and unit conversions. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Two blue and three gold. That set is the ratio 2 to 3.
A ratio is a set you keep taking. 2:3 and 6:9 are the SAME ratio, because 6:9 is that same set three times. To keep a ratio the same you MULTIPLY both parts by the same number — never add.
What people get wrong
⚠️People often think…
A ratio of 2:3 means there are 3 more gold than blue.
A ratio compares by GROUPS, not by difference. 2:3 means two blue for every three gold — which could be 2 and 3, or 20 and 30. The gap changes; the ratio does not.
Groups, not the gap between them.
⚠️People often think…
To keep a ratio the same, add the same number to both sides.
Adding changes it. 2:3 plus one each is 3:4, which is a different mix entirely — more blue per gold than before. Multiplying preserves it: 2:3 doubled is 4:6, the same mix.
Multiply to keep it. Adding changes it.
⚠️People often think…
For a unit rate, divide the bigger number by the smaller one.
Divide by whatever you want ONE of. “Miles per hour” means one hour, so divide miles by hours — even if hours is the bigger number. The word “per” tells you the denominator.
The word “per” names the denominator.
⚠️People often think…
If blue to gold is 2:3, then 2/3 of them are blue.
A ratio compares the two PARTS; a fraction compares a part to the WHOLE. The whole is 2 + 3 = 5, so blue is 2/5 of them. Add the parts before you write a fraction.
Add the parts to get the whole.
Same set, more times
1Watch one
A recipe uses 2 cups flour to 3 cups milk. You want 9 cups of milk. How much flour?
Find how many times bigger: 9 ÷ 3 = 3.
Whatever you did to one part, do to the other — multiply, not add.
2 × 3 = 6 cups of flour.
Check: 6:9 is the same mix as 2:3. Adding 6 to each would have given 8:9, which tastes wrong. ✓
2Do one with me
Blue to gold is 2:3, and a car goes 120 miles in 3 hours.
The fraction of the whole that is blue is
To keep a ratio the same you
The car’s speed in miles per hour is
💬One sentence, then you move on
Why does adding to both sides change a ratio?
3Try one
A store sells 5 pounds for $12. What is the price per pound?
I want a hint first
You want the cost of ONE pound, so pounds goes on the bottom: 12 ÷ 5.
💬Last one — then you're done here
Why is 2:3 not the same as 2/3 of the whole?
Where this goes
Where this lives
Doubling a recipe, comparing two package sizes at a store, working out a speed from a distance and a time.
What this feeds
Next unit is percent and proportional relationships — ratios written a different way.
Name one place you compare two amounts per one of something.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 From a comparison to a unit rate — the ladder
Ratio (for every) → equivalent ratios (multiply both) → ratio table → rate (two different units) → unit rate (for ONE) → use it: multiply to go up, divide to come back.
Compare, scale, divide to one, multiply to many.
🤝 The whole topic in one idea
Whatever you do to one number of a ratio, do the same MULTIPLYING or DIVIDING to the other. A table, a tape diagram and a double number line are just three ways of keeping track of that one move.
⚠️ Traps the test loves
Order. “Dogs to cats” and “cats to dogs” are different ratios. Match the words.
Part or whole? “Red to blue” is part to part. “Red to all” needs the total — add first.
Adding is not scaling. 2:3 plus 1 each is 3:4, a different ratio.
Unit price is dollars ÷ units. Flipping it gives units per dollar — then HIGHER is better.
The cheaper package is not always the better buy. Compare unit prices, not totals.
“30% of what is 12?” is not “30% of 12.” Decide which number is the whole.
📐 Four pictures of one ratio — the chart
Tool
What it looks like
Best for
Ratio table
rows of equivalent ratios
jumping to any amount — multiply a row
Tape diagram
equal boxes for each part
same-unit ratios with a total or a difference
Double number line
two lines, zeros lined up
rates and percents — two different units
Graph
points on a line through (0, 0)
seeing at a glance which row does not belong
🎯 How the test will ask
A picture of objects — “write the ratio of ___ to ___” (watch the order, and part vs whole).
A table with a blank — find the multiplier from a full row, then use it.
Two packages — “which is the better buy?” (two divisions, then compare).
A trip — distance, rate, time: two are given, find the third.
A percent — find the part (30% of 50) or the whole (30% of what is 15?).
A unit change — feet to inches, minutes to hours. Smaller unit, bigger number.
✅ Before the test, can you…
Write a ratio three ways, in the order the words ask for?
Fill in a ratio table and spot the row that does not belong?
Solve a “ratio and a total” problem with a tape diagram?
Find a unit rate and use it to go up to any amount?
Pick the better buy and show the two unit prices?
Find a percent of a number, and the whole when you know a part and its percent?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: fill in the number line, sort it, graph it, then argue from the numbers. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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The answer key is for a teacher: it prints only from here, for the unit on screen. Print the study packet prints the study pages and a blank quiz — never the answers.